Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{5})^{2}}}{{(a^{-5}z^{2})^{-1}}}$
Answer: To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{5}}$ to the exponent ${2}$ . Now ${5 \times 2 = 10}$ , so ${(a^{5})^{2} = a^{10}}$ In the denominator, we can use the distributive property of exponents. ${(a^{-5}z^{2})^{-1} = (a^{-5})^{-1}(z^{2})^{-1}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{5})^{2}}}{{(a^{-5}z^{2})^{-1}}} = \dfrac{{a^{10}}}{{a^{5}z^{-2}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{10}}}{{a^{5}z^{-2}}} = \dfrac{{a^{10}}}{{a^{5}}} \cdot \dfrac{{1}}{{z^{-2}}} = a^{{10} - {5}} \cdot z^{- {(-2)}} = a^{5}z^{2}$.